# Semi-log plot

__: Type of graph__

**Short description**In science and engineering, a **semi-log plot**/**graph** or **semi-logarithmic** **plot**/**graph** has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of values, or to zoom in and visualize that - what seems to be a straight line in the beginning - is in fact the slow start of a logarithmic curve that is about to spike and changes are much bigger than thought initially.^{[1]}

All equations of the form [math]\displaystyle{ y=\lambda a^{\gamma x} }[/math] form straight lines when plotted semi-logarithmically, since taking logs of both sides gives

- [math]\displaystyle{ \log_a y = \gamma x + \log_a \lambda. }[/math]

This is a line with slope [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ \log_a \lambda }[/math] vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2:

- [math]\displaystyle{ \log (y) = (\gamma \log (a)) x + \log (\lambda). }[/math]

A **log–linear** (sometimes log–lin) plot has the logarithmic scale on the *y*-axis, and a linear scale on the *x*-axis; a **linear-log** (sometimes lin–log) is the opposite. The naming is *output-input* (*y*-*x*), the opposite order from (*x*, *y*).

On a semi-log plot the spacing of the scale on the *y*-axis (or *x*-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the *y* values (or *x* values) to their log, and plotting the data on linear scales. A log–log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.

## Equations

The equation of a line on a log–linear plot, where the abscissa axis is scaled logarithmically (with a logarithmic base of *n*), would be

- [math]\displaystyle{ F(x) = m \log_{n}(x) + b. \, }[/math]

The equation for a line on a linear–log plot, with an ordinate axis logarithmically scaled (with a logarithmic base of *n*), would be:

- [math]\displaystyle{ \log_{n}(F(x)) = mx + b }[/math]
- [math]\displaystyle{ F(x) = n^{mx + b} = (n^{mx})(n^b). }[/math]

### Finding the function from the semi–log plot

#### Linear-log plot

On a linear-log plot, pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) on the same graph. The slope formula of the plot is:

- [math]\displaystyle{ m = \frac {F_1 - F_0}{\log_n (x_1 / x_0)} }[/math]

which leads to

- [math]\displaystyle{ F_1 - F_0 = m \log_n (x_1 / x_0) }[/math]

or

- [math]\displaystyle{ F_1 = m \log_n (x_1 / x_0) + F_0 = m \log_n (x_1) - m \log_n (x_0) + F_0 }[/math]

which means that

- [math]\displaystyle{ F(x) = m \log_n (x) + constant }[/math]

In other words, *F* is proportional to the logarithm of *x* times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points (*F*_{0}, *x*_{0}) and (*F*_{1}, *x*_{1}) will have the function:

- [math]\displaystyle{ F(x) = (F_1 - F_0) {\left[\frac{\log_n (x / x_0)}{\log_n(x_1 / x_0)}\right]} + F_0 = (F_1 - F_0) \log_{\frac{x_1}{x_0}}{\left(\frac{x}{x_0}\right)} + F_0 }[/math]

#### log–linear plot

On a log–linear plot (logarithmic scale on the y-axis), pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) on the same graph. The slope formula of the plot is:

- [math]\displaystyle{ m = \frac {\log_n (F_1 / F_0)}{x_1 - x_0} }[/math]

which leads to

- [math]\displaystyle{ \log_n(F_1 / F_0) = m (x_1 - x_0) }[/math]

Notice that *n*^{logn(F1)} = *F*_{1}. Therefore, the logs can be inverted to find:

- [math]\displaystyle{ \frac{F_1}{F_0} = n^{m(x_1 - x_0)} }[/math]

or

- [math]\displaystyle{ F_1 = F_0n^{m(x_1 - x_0)} }[/math]

This can be generalized for any point, instead of just *F _{1}*:

- [math]\displaystyle{ F(x) = {F_0} n^{\left(\frac {x-x_0}{x_1-x_0}\right) \log_n (F_1 / F_0)} }[/math]

## Real-world examples

### Phase diagram of water

In physics and chemistry, a plot of logarithm of pressure against temperature can be used to illustrate the various phases of a substance, as in the following for water:

### 2009 "swine flu" progression

While ten is the most common base, there are times when other bases are more appropriate, as in this example:

### Microbial growth

In biology and biological engineering, the change in numbers of microbes due to asexual reproduction and nutrient exhaustion is commonly illustrated by a semi-log plot. Time is usually the independent axis, with the logarithm of the number or mass of bacteria or other microbe as the dependent variable. This forms a plot with four distinct phases, as shown below.

thumb|none|500px|Bacterial growth curve

## See also

- Nomograph, more complicated graphs
- Nonlinear regression, for converting a nonlinear form to a semi-log form amenable to non-iterative calculation
- Log–log plot

## References

- ↑ (1) Bourne, M.. "Graphs on Logarithmic and Semi-Logarithmic Paper".
*Interactive Mathematics*. www.intmath.com. http://www.intmath.com/Exponential-logarithmic-functions/7_Graphs-log-semilog.php.

(2) Bourne, Murray (January 25, 2007). "Interesting semi-logarithmic graph - YouTube Traffic Rank".*SquareCirclez: The IntMath blog*. www.intmath.com. https://www.intmath.com/blog/mathematics/interesting-semi-logarithmic-graph-youtube-traffic-rank-526.

Original source: https://en.wikipedia.org/wiki/ Semi-log plot.
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